On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type

We consider the finite W-algebra U(g, e) associated to a nilpotent element e is an element of g in a simple complex Lie algebra g of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem for U(g, e), we verify a conjecture of Premet, that U (g, e) always has a 1-dimensional representation when g is of type G(2), F-4, E-6 or E-7. Thanks to a theorem of Premet,this allows one to deduce the existence of minimal dimension representations of reduced enveloping algebras of modular Lie algebras of the above types. In addition, a theorem of Losev allows us to deduce that there exists a completely prime primitive ideal in U(g) whose associated variety is the coadjoint orbit corresponding to e.